Of the myriad ways in which a rock band can crash and burn—getting caught up in arguments over who wrote what, signing a slave-for-life contract, developing a fondness for the pampered life—one of the surest is to begin to think itself important.

The Clash, that mighty roar of a rock band that was so much more, came perilously close to falling victim to its own press after it released its eponymous debut album 30 years ago this month. Heralded as “the most important rock band in the world,” the group tore through the late 1970s as if on some secret assignment to change history. Tours became crusades, ordinary one-off shows life-or-death performances, all to the benefit of fans lucky enough to catch them live. Songs such as “I’m So Bored with U.S.A.” and “White Man in Hammersmith Palais” spoke to disaffected British youth in a time of supposed jubilee, and then spread out across the world, the lyrics not always understood, the tunes grokked instantly.

Then came the party line of the 1980s: the group’s members, no longer mates, accused one another of selling out, squabbled over political differences as some came to prefer limos to Doc Martens, and broke up noisily and nastily.

For most of its nine-year run made up of guitarists Joe Strummer and Mick Jones, bassist Paul Simonon, and drummer Nicky Headon, The Clash was important. They helped turn the dinosaurish arena rock scene on its head, making the fat-and-happy likes of Rod Stewart and Pink Floyd seem completely irrelevant. (Some, such as the Floyd, returned from the ranks of the forgettable. Others did not.)

Wedding reggae, rock, jazz, and other musical traditions, The Clash served up hearty, beat-heavy dissent. Strummer and company treated authority figures of whatever ilk, from Mao to Reagan, with equal disdain, but they offered a vision substantially more profound than the live-fast-die-young credo of many groups of the time: if you don’t like the world, they urged, then change it. The Clash mattered, in short, as few other bands ever have, a case that former Mojo editor Pat Gilbert capably makes in his recent book Passion Is a Fashion.

The four were basically nice middle-class kids who had to be schooled in what Gilbert calls “the speedy, yobbish milieu that would become known as punk.” When the roots revolution began, after all, they were still digging Slade and ABBA. When it ended, they had conquered the world—or the equivalent, having scored a Top 10 U.S. hit in 1982 with “Rock the Casbah,” a song, for reasons that aren’t entirely obvious, reported to be much favored by British and American troops in Iraq today.

They had also done most of what they swore they never would, adopting rock-star ways and removing old friends from their inner circle because they didn’t fit in with the fashion of the hour. “There was always a conflict,” a confidant recalled, “between them being a rock band and being a revolutionary outfit.”

Indeed. After hitting the charts, Joe Strummer complained that the only thing they could possibly do now was to become the rich rock stars they had pledged to destroy. Yet they stayed true to a vision that treated the fans as participants, not wallets to be emptied, insisting that albums such as the two-disc London Calling and the three-disc Sandinista! be priced only slightly higher than single albums. The record-company executives were unhappy, but they gave in, and the band stayed broke, a badge of rock ‘n’ roll honor.

The money came all the same. And while The Clash didn’t have all the answers and took more than a few wrong turns, the band knew how to make people dance—and dance a rebel waltz at that. The Clash were important, and a great rock band, full stop. Thirty years on, they remain so: fire up “Complete Control” or any other cut from that debut album, and see for yourself.